On the quasiarithmetic mean in a mean value property and the associated functional equation

Summary

The functional equation

$$\frac{{xf(y) - yf(x)}}{{x - y}} = \varphi [\zeta (x,y)], x,y \in I, x \ne y,$$

associated with a mean value property, is considered. HereI is a real interval of positive length and ζ(x, y) is a quasiarithmetic mean ofx andy. In the particular case when 0 εI, or when 0 ∉I and ζ is the arithmetic, geometric, or harmonic mean, equation (*) has been solved previously by J. Aczél and the author. Now the general case is dealt with.

The general solution of equation (*) is described in the case where ζ is a quasiarithmetic mean. No regularity assumptions are made. The method is illustrated by examples. In particular, the earlier results of J. Aczél and the author concerning equation (*) are obtained here again as consequences of the general theorem.